Starting out with batteries that spec out at 1.280 on all cells and diligently recharging to 90% capacity after discharging to 50%, you can go 4 weeks. But try one day in the middle of your stay to reach 95-98% capacity with the Black & Decker. Four weeks is stretching things a bit far even with 90% recharging. I would not want to be forced to do an equalization upon returning home and that 95-98% recharge at the halfway point will make a big difference.
But having the generator running that long on day 14 may cramp your style of camping.
